1,318 research outputs found
Partially-commutative context-free languages
The paper is about a class of languages that extends context-free languages
(CFL) and is stable under shuffle. Specifically, we investigate the class of
partially-commutative context-free languages (PCCFL), where non-terminal
symbols are commutative according to a binary independence relation, very much
like in trace theory. The class has been recently proposed as a robust class
subsuming CFL and commutative CFL. This paper surveys properties of PCCFL. We
identify a natural corresponding automaton model: stateless multi-pushdown
automata. We show stability of the class under natural operations, including
homomorphic images and shuffle. Finally, we relate expressiveness of PCCFL to
two other relevant classes: CFL extended with shuffle and trace-closures of
CFL. Among technical contributions of the paper are pumping lemmas, as an
elegant completion of known pumping properties of regular languages, CFL and
commutative CFL.Comment: In Proceedings EXPRESS/SOS 2012, arXiv:1208.244
Commutative positive varieties of languages
We study the commutative positive varieties of languages closed under various
operations: shuffle, renaming and product over one-letter alphabets
Cumulants, free cumulants and half-shuffles
Free cumulants were introduced as the proper analog of classical cumulants in
the theory of free probability. There is a mix of similarities and differences,
when one considers the two families of cumulants. Whereas the combinatorics of
classical cumulants is well expressed in terms of set partitions, the one of
free cumulants is described, and often introduced in terms of non-crossing set
partitions. The formal series approach to classical and free cumulants also
largely differ. It is the purpose of the present article to put forward a
different approach to these phenomena. Namely, we show that cumulants, whether
classical or free, can be understood in terms of the algebra and combinatorics
underlying commutative as well as non-commutative (half-)shuffles and
(half-)unshuffles. As a corollary, cumulants and free cumulants can be
characterized through linear fixed point equations. We study the exponential
solutions of these linear fixed point equations, which display well the
commutative, respectively non-commutative, character of classical, respectively
free, cumulants.Comment: updated and revised version; accepted for publication in PRS
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